Alexander Bass
This note is one of many taken during 2024

Integrals of Step Functions

The integral of the function ⌊x⌋\lfloor x \rfloor with respect to xx is equal to

∫⌊x⌋ dx==∑n=0⌊x⌋−1n+(x−⌊x⌋)(⌊x⌋)=12(⌊x⌋−1)(⌊x⌋−1+1)+(x−⌊x⌋)⌊x⌋=(x−12)⌊x⌋−12⌊x⌋2 \begin{align*} \int \lfloor x \rfloor \, dx =& \\ =& \sum_{n=0}^{\lfloor x \rfloor -1} n + (x-\lfloor x \rfloor )(\lfloor x \rfloor )\\ =& \frac{1 }{2} (\lfloor x \rfloor -1)(\lfloor x \rfloor -1 +1 ) + (x-\lfloor x \rfloor )\lfloor x \rfloor \\ =& \left( x-\frac{1}{2} \right) \lfloor x \rfloor - \frac{1}{2} \lfloor x \rfloor^2 \end{align*}

(Constant of integration omitted)

A similar process can be done for the ceiling function ⌈x⌉\lceil x \rceil.

∫⌈x⌉ dx==∑n=0⌈x⌉n+(x−⌈(x)⌉)⌈x⌉=12⌈x⌉(⌈x⌉+1)+(x−⌈x⌉)⌈x⌉=(x+12)⌈x⌉−12⌈x⌉2 \begin{align*} \int \lceil x \rceil \, dx =& \\ =&\sum_{n=0}^{\lceil x \rceil } n +(x-\lceil (x) \rceil )\lceil x \rceil \\ =& \frac{1}{2} \lceil x \rceil(\lceil x \rceil +1 ) + (x-\lceil x \rceil )\lceil x \rceil \\ =& \left( x+\frac{1}{2} \right)\lceil x \rceil -\frac{1}{2} \lceil x \rceil^2 \end{align*}

I’ll spare the fine details on the final one. The ‘round to the nearest integer’ function nint⁡(x)\operatorname{nint}(x) can also be integrated.

∫nint⁡(x) dx=x(nint⁡(x))−12nint⁡(x)2 \int \operatorname{nint}(x) \, dx = x(\operatorname{nint} (x)) - \frac{1}{2} \operatorname{nint}(x)^2
This note is one of many taken during 2024